| 31. | This is the tactic taken by the covariant derivative approach to connections : good behavior is equated with covariance.
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| 32. | Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.
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| 33. | By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.
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| 34. | In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves.
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| 35. | Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative.
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| 36. | With the classical covariant derivatives, covariance is an " a posteriori " feature of the derivative.
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| 37. | Then decompose the ambient covariant derivative of ? along " X " into tangential and normal components:
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| 38. | Where we have taken advantage of the first covariant derivative of a function being the same as its ordinary derivative.
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| 39. | Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport.
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| 40. | Furthermore, many of the features of the covariant derivative still remain : parallel transport, curvature, and holonomy.
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